reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;
reserve p,q for Polynomial of F_Real;

theorem Th46:
  for p being constant Polynomial of F_Real holds poly_diff(p) = 0_.F_Real
  proof
    let p be constant Polynomial of F_Real;
    per cases;
    suppose
      p <> z;
      then
A1:   len p = len poly_diff(p) + 1 by Th45;
      deg p <= 0 by RATFUNC1:def 2;
      then len poly_diff(p) = 0 by A1;
      then deg poly_diff(p) = -1;
      hence thesis by HURWITZ:20;
    end;
    suppose
A2:   p = z;
      let n be Element of NAT;
      thus z.n = z.(n+1) * (n+1)
      .= (poly_diff(p)).n by A2,Def5;
    end;
  end;
